Purchase train tickets at the Train Station - Roma Termini is the largest train station in Rome - you can also purchase train tickets at tobacco shops, coffee bars, etc.

Your ticket will look something like this (note, if you take the Leonardo express from the airport, you may not have a seat assignment since it is analogous to catching the link light rail from Seatac):

Validate your ticket at the train tracks by stamping it using the yellow stamp box (think of it like the orca box for the link light rail that you tap before riding the train – except the one in Italy actually stamps a date and time onto your ticket)